Lemma 8 [1]}. Let \(G\)  be a connected graph and \(D(G)\)  be the distance matrix of \(G\) . Then \(\lambda _n(D(G))=-2\)  with multiplicity \(n-k\)  if and only if \(G\)  is a complete \(k\) -partite graph for \(2\le k\le n-1\) .

